Deriving neural scaling laws from the statistics of natural language

A paper posted to arXiv on 7 Feb 2026 by Francesco Cagnetta claims the first theory that quantitatively predicts data-limited neural scaling exponents for modern LLMs trained on natural language. The theory isolates two statistical properties of language — the decay of pairwise token correlations with time separation and the decay of next-token conditional entropy with conditioning-context length — and derives a parameter-free formula that matched experimentally measured scaling laws from GPT-2 and LLaMA style models trained from scratch on TinyStories and WikiText.

Computer Science Machine Learning Submitted on 7 Feb 2026 v1 https://arxiv.org/abs/2602.07488v1 , last revised 3 Jul 2026 this version, v3 Title:Deriving Neural Scaling Laws from the statistics of natural language View PDF https://arxiv.org/pdf/2602.07488 HTML experimental https://arxiv.org/html/2602.07488v3 Abstract:Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of language that alone can predict neural scaling exponents: i the decay of pairwise token correlations with time separation between token pairs, and ii the decay of the next-token conditional entropy with the length of the conditioning context. We further derive a simple formula in terms of these statistics that predicts data-limited neural scaling exponents from first principles without any free parameters or synthetic data models. Our theory exhibits a remarkable match with experimentally measured neural scaling laws obtained from training GPT-2 and LLaMA style models from scratch on two qualitatively different benchmarks, TinyStories and WikiText. Submission history From: Francesco Cagnetta view email https://arxiv.org/show-email/c6400098/2602.07488 Sat, 7 Feb 2026 10:40:28 UTC 2,072 KB \ v1\ https://arxiv.org/abs/2602.07488v1 Thu, 12 Feb 2026 11:54:22 UTC 2,072 KB \ v2\ https://arxiv.org/abs/2602.07488v2 v3 Fri, 3 Jul 2026 02:02:48 UTC 3,083 KB Current browse context: cs.LG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender IArxiv Recommender What is IArxiv? https://iarxiv.org/about arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .